I would like to better understand a certain compactification of the Jacobian variety of a singular algebraic plane curve as described in Cook's Ph.D. 1993 thesis Local and Global Aspects of the Module Theory of Singular Curves. So far, I have only been able to find references discussing this compactification for the case of simple/ADE singularities (e.g. [1] [2]).

Question: Is there some kind of obstruction and/or limitation of our current tools that prevents this compactification from working for arbitrary singularities? What parts of the aforementioned construction in Cook's thesis crucially rely on the assumption that every singularity of the curve is simple?

[1] Emilio Franco, Robert Hanson, Johannes Horn, André Oliveira, Fourier-Mukai transforms and normalization of nodal curves, https://arxiv.org/abs/2405.11860
[2] Peter B. Gothen, André Oliveira, The singular fibre of the Hitchin map, Int. Math. Res. Notices 2013 no. 5 (2013), 1079-1121. https://doi.org/10.1093/imrn/rns020. https://arxiv.org/abs/1012.5541